Games of No Chance 4
MSRI Publications
Volume 63, 2015
Unsolved problems in combinatorial games
RICHARD J. NOWAKOWSKI
We have sorted the problems into sections:
•
A. Taking and breaking
•
B. Pushing and placing pieces
•
C. Playing with pencil and paper
•
D. Disturbing and destroying
•
E. Theory of games
The numbers in parentheses are the old numbers used in each of the lists of
unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math.
43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance,
hereafter referred to as GONC; on pp. 457–473 of More Games of No Chance
(MGONC); and on pp. 475–500 of Games of No Chance 3 (GONC3). Some
numbers have little more than the statement of the problem if there is nothing
new to be added. References [year] may be found in Fraenkel’s bibliography
at the end of this volume. References [#] are at the end of this article. A
useful reference for the rules and an introduction to many of the specific games
mentioned below is M. Albert, R. J. Nowakowski and D. Wolfe, Lessons in Play:
An Introduction to the Combinatorial Theory of Games, A. K. Peters, 2007 (LIP)
or Berlekamp, Conway and Guy, Winning Ways for your Mathematical Plays,
vol. 1–4, A. K. Peters, 2000–2004 (WW).
A. Taking and breaking games
A1. (1) Subtraction games with finite subtraction sets are known to have periodic nim-sequences. Investigate the relationship between the subtraction set
and the length and structure of the period. The same question can be asked
about partizan subtraction games, in which each player is assigned an individual
subtraction set. See Fraenkel and Kotzig [1987].
(A move in the game S(s1 , s2 , s3 , . . .) is to take a number of beans from a
heap, provided that number is a member of the subtraction-set, {s1 , s2 , s3 , . . .}.
Keywords: none.
279
280
RICHARD J. NOWAKOWSKI
Analysis of such a game and of many other heap games is conveniently recorded
by a nim-sequence,
n0n1n2n3 . . . ,
meaning that the nim-value of a heap of h beans is n h ; i.e., that the value of a
heap of h beans in this particular game is the nimber ∗n h .)
For examples see Table 2 in Section 4 on p. 67 of “Impartial games” in GONC.
It would now seem feasible to give the complete analysis for games whose
subtraction sets have just three members, though this has so far eluded us. Several
people, including Mark Paulhus and Alex Fink, have given a complete analysis
for all sets {1, b, c} and for sets {a, b, c} with a < b < c < 32.
In general, period lengths can be surprisingly long, and it has been suggested
that they could be superpolynomial in terms of the size of the subtraction set.
However, Guy conjectures that they are bounded by polynomials of degree
at most n2 in sn , the largest member of a subtraction set of cardinality n. It
would also be of interest to characterize the subtraction sets which yield a purely
periodic nim-sequence, i.e., for which there is no preperiod. Carlos Santos [6]
reduced this upper bound slightly by using a dynamical system approach.
Angela Siegel [20] considered infinite subtraction sets which are the complement of finite ones and showed that the nim-sequences are always arithmetic
periodic. That is, the nim-values belong to a finite set of arithmetic progressions with the same common difference. The number of progressions is the
period and their common difference is called the saltus. For instance, the game
b 30}
b (in which a player may take any number of beans except 4, 9,
S{4̂, 9̂, 26,
26 or 30) has a preperiod of length 243, period-length 13014 and saltus 4702.
Marla Clusky and Danny Sleator have proved her conjecture that the class of
games S{â, b̂, a[
+ b} is purely periodic with period length 3(a + b).
For infinite subtraction games in general there are corresponding questions
about the length and purity of the period. Suppose the elements of the finite
subtraction sets are not constants. It was shown by Fraenkel [2011] that the game
with subtraction set
1, 2, . . . , t − 1, bn/tc ,
where t ≥ 2 is a fixed integer, n the pile size, has an aperiodic Sprague–Grundy
sequence. See also Fraenkel [2012] and Guo [2012].
A2. (2) Are all finite octal games ultimately periodic? (If the binary expansion
of the k-th code digit in the game with code d0 ·d1 d2 d3 . . . is
dk = 2ak + 2bk + 2ck + · · · ,
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
281
Figure 1. Plot of 11000000 nim-values of the octal game ·007.
where 0 ≤ ak < bk < ck < . . . , then it is legal to remove k beans from a heap,
provided that the rest of the heap is left in exactly ak or bk or ck or . . . nonempty
heaps. See WW, pp. 81–115. Some specimen games are exhibited in Table 3 of
Section 5 of “Impartial games” in GONC.)
Resolve any number of outstanding particular cases, e.g., ·6 (Officers), ·04,
·06, ·14, ·36, ·37, ·64, ·74, ·76, ·004, ·005, ·006, ·007, ·014, ·015, ·016, ·024,
·026, ·034, ·064, ·114, ·125, ·126, ·135, ·136, ·142, ·143, ·146, ·162, ·163, ·164,
·166, ·167, ·172, ·174, ·204, ·205, ·206, ·207, ·224, ·244, ·245, ·264, ·324, ·334,
·336, ·342, ·344, ·346, ·362, ·364, ·366, ·371, ·374, ·404, ·414, ·416, ·444, ·564,
·604, ·606, ·744, ·764, ·774, ·776 and Grundy’s Game (split a heap into two
unequal heaps; WW, pp. 96–97, 111–112; LIP, p. 142), which has been analyzed,
first by Dan Hoey, and later by Achim Flammenkamp, as far as heaps of 235
beans.
J.P. Grossman (preprint) using a new approach based on the sparse space
phenomenon, has analyzed ·.6 up to heaps of size 247 and has found no periodicity.
Perhaps the most notorious and deserving of attention is the game ·007, onedimensional Tic-Tac-Toe, or Treblecross, which Flammenkamp has pushed to
225 . Figure 1 shows the first 11 million nim-values, a small proportion of which
are ≥ 1024; the largest, G(6193903) = 1401 is shown circled. Will 2048 ever be
reached?
Flammenkamp has settled ·106: it has the remarkable period and preperiod
lengths of 328226140474 and 465384263797. For information on the current
status of each of these games, we refer the read to Flammenkamp’s web page at
uni-bielefeld.de/ achim/octal.html.
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RICHARD J. NOWAKOWSKI
A game similar to Grundy’s, and which is also unsolved, is John Conway’s
Couples-Are-Forever (LIP, p. 142) where a move is to split any heap except a
heap of two. The first 50 million nim-values haven’t displayed any periodicity.
See Caines et al. [1999]. More generally, Bill Pulleyblank suggests looking at
splitting games in which you may only split heaps of size > h, so that h = 1 is
She-Loves-Me-She-Loves-Me-Not and h = 2 is Couples-Are-Forever. David
Singmaster suggested a similar generalization: you may split a heap provided
that the resulting two heaps each contain at least k beans: k = 1 is the same as
h = 1, while k = 2 is the third cousin of Dawson’s Chess.
Explain the structure of the periods of games known to be periodic.
In Discrete Math., 44 (1983), pp. 331–334, Problem 38, Fraenkel raised
questions concerning the computational complexity (see Section E1 below) of
octal games. In Problem 39, he and Kotzig define partizan octal games in which
distinct octals are assigned to the two players. Mesdal [2009] show that in many
cases, if the game is “all-small” (WW, pp. 229–262; LIP, pp. 183–207), then the
atomic weights are arithmetic periodic. (See Section E13 for an explanation of
the new name “dicot” games.) In Problem 40, Fraenkel introduces poset games,
played on a partially ordered set of heaps, each player in turn selecting a heap
and then removing a nonnegative number of beans from this heap and from each
heap above it in the ordering, at least one heap being reduced in size. For posets
of height one, new regularities in the nim-sequence can occur; see Horrocks and
Nowakowski [2003].
Note that this includes, as particular cases, Subset Takeaway, Chomp or
Divisors, and Green Hackenbush forests. Compare Problems A3, D1 and D2
below.
A3. (3) Hexadecimal games have code digits dk in the interval from 0 to
f (= 15), so that there are options splitting a heap into three heaps. See WW,
pp. 116–117.
Such games may be arithmetically periodic. Nowakowski has calculated
the first 100000 nim-values for each of the 1-, 2- and 3-digit games. Richard
Austin’s Theorem 6.8 in his thesis [1976] and the generalization by Howse and
Nowakowski [2004] suffice to confirm the arithmetic periodicity of several of
these games.
Some interesting specimens are ·28 = ·29, which have period 53 and saltus 16,
the only exceptional value being G(0) = 0; ·9c, which has period 36, preperiod
28 and saltus 16; and ·f6 with period 43 and saltus 32, but its apparent preperiod
of 604 and failure to satisfy one of the conditions of the theorem prevent us from
verifying the ultimate periodicity. The game ·205200c is arithmetic periodic with
preperiod length of 4, period length of 40, saltus 16 except that 40k + 19 has
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
283
nim-value 6 and 40k + 39 has nim-value 14. This regularity, (which also seems
to be exhibited by ·660060008 with a period length of approximately 300,000),
was first reported in Horrocks and Nowakowski [2003] (see Problem A2). Grossman and Nowakowski [this volume] have shown that the nim-sequences for
·200. . . 0048, with an odd number of zero code digits, exhibit “ruler function”
patterns. The game ·9 has not so far yielded its complete analysis, but, as far
as analyzed (to heaps of size 100000), exhibits a remarkable fractal-like set of
nim-values. See Howse and Nowakowski [2004]. Also of special interest are ·e;
·7f (which has a strong tendency to period 8, saltus 4, but, for n ≤ 100, 000, has
14 exceptional values, the largest being G(94156) = 26614); ·b6 (which “looks
octal”); ·b33b (where a heap of size n has nim-value n except for 27 heap sizes
which appear to be random).
Other unsolved hexadecimal games are
·1x,
·3x,
·5x,
·7x,
·9d,
·dx,
x ∈ {8, 9, c, d, e, f },
8 ≤ x ≤ e,
8 ≤ x ≤ f,
8 ≤ x ≤ f,
1 ≤ x ≤ f,
·2x,
·4x,
·6x,
·9x,
·bx,
·fx,
a ≤ x ≤ f,
x ∈ {9, b, d, f },
8 ≤ x ≤ f,
1 ≤ x ≤ a,
x ∈ {6, 9, d},
x ∈ {4, 6, 7}.
A4. (53) N-heap Wythoff game. Given N ≥ 2 heaps of finitely many tokens,
whose sizes are p1 , . . . , p N with p1 ≤ · · · ≤ p N . Players take turns removing any
positive number of tokens from a single heap or removing (a1 , . . . , a N ) from all
the heaps — ai from the i-th heap — subject to the conditions: (i) 0 ≤ ai ≤ pi for
PN
each i, (ii) i=1
ai > 0, (iii) a1 ⊕ · · · ⊕ a N = 0, where ⊕ is nim addition. The
player making the last move wins and the opponent loses. Note that the classical
Wythoff game is the case N = 2.
For N ≥ 3, Fraenkel makes the following conjectures.
Conjecture 1. For every fixed set K := (A1 , . . . , A N −2 ) there exists an integer
m = m(K ) (i.e, m depends only on K ), such that
A1 , . . . , A N −2 , AnN −1 , AnN , A N −2 ≤ AnN −1 ≤ AnN ,
N −1
with AnN −1 < An+1
for all n ≥ 1, is the n-th P-position, and
AnN −1 = mex {AiN −1 , AiN : 0 ≤ i < n} ∪ T , AnN = AnN −1 + n,
for all n ≥ m, where T = T (K ) is a (small) set of integers.
That is, if you fix N − 2 of the heaps, the P-positions resemble those for
the classical Wythoff game. For example, for N = 3 and A1 = 1, we have
T = {2, 17, 22}, m = 23.
284
RICHARD J. NOWAKOWSKI
Conjecture 2. For every fixed K there exist integers a = a(K ) and M = M(K )
such that
AnN −1 = bnφc + εn + a and AnN = AnN −1 + n,
√
for all n ≥ M, where φ = (1 + 5)/2 is the golden section, and εn ∈ {−1, 0, 1}.
In Fraenkel and Krieger
√ [2004] the following was shown, inter alia: let
t ∈ Z≥1 , α = (2 − t + t 2 + 4)/2 (α = φ for t = 1), T ⊂ Z≥0 a finite set,
An = (mex {Ai , Bi : 0 ≤ i < n} ∪ T ), where Bn = An + nt. Let sn := bnαc − An .
Then there exist a ∈ Z and m ∈ Z≥1 , such that for all n ≥ m, either sn = a, or
sn = a + εn , εn ∈ {−1, 0, 1}. If εn 6= 0, then εn−1 = εn+1 = 0. Also the general
structure of the εn was characterized succinctly.
This result was then applied to the N -heap Wythoff game. In particular, for
N = 3 (such that K = A1 ) it was proved that A2n = mex {Ai2 , Ai3 : 0 ≤ i < n}∪T ,
where
T = {x ≥ K : there exists 0 ≤ k < K such that (k, K , x) is a P-position}
∪{0, . . . , K − 1}.
The following upper bound for A3n was established: A3n ≤ (K + 3)A2n + 2K + 2.
It was also proved that Conjecture 1 implies Conjecture 2.
In Sun and Zeilberger [2004], a sufficient condition for the conjectures to hold
was given. It was then proved that the conjectures are true for the case N = 3,
where the first heap has up to 10 tokens. For those 10 cases, the parameter values
m, M, a, T were listed in a table.
Sun [2005] obtained results similar to those in Fraenkel and Krieger [2004], but
the proofs are different. It was also proved that Conjecture 1 implies Conjecture 2.
A method was given to compute a in terms of certain indexes of the Ai and B j .
A5. (23) Burning-the-Candle-at-Both-Ends. Conway and Fraenkel ask us to
analyze Nim played with a row of heaps. A move may only be made in the
leftmost or in the rightmost heap. When a heap becomes empty, then its neighbor
becomes the end heap.
Albert and Nowakowski [2001] have determined the outcome classes in impartial and partizan versions (called End-Nim, LIP, pp. 210, 263) with finite
heaps, and Duffy, Kolpin and Wolfe [2009] extend the partizan case to infinite
ordinal heaps. Wolfe asks for the actual values.
Nowakowski suggested to analyze impartial and partizan End-Wythoff: take
from either end-pile, or the same number from both ends. The impartial game is
solved by Fraenkel and Reisner [2009]. Fraenkel [1982] asks a similar question
about a generalized Wythoff game: take from either end-pile or take k > 0 from
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
285
one end-pile and ` > 0 from the other, subject to |k − `| < a, where a is a fixed
integer parameter (a = 1 is End-Wythoff).
There is also Hub-and-Spoke Nim, proposed by Fraenkel. One heap is the
hub and the others are arranged in rows forming spokes radiating from the hub.
Albert notes that this game can be generalized to playing on a forest, i.e., a graph
each of whose components is a tree. The most natural variant is that beans may
only be taken from a leaf (valence 1) or isolated vertex (valence 0).
The partizan game of Red-Blue Cherries is played on an arbitrary graph. A
player picks an appropriately colored cherry from a vertex of minimum degree,
which disappears at the same time. Albert, Grossman, McCurdy, Nowakowski
and Wolfe [1] show that if the graph has a leaf, then the value is an integer. They
ask: Is every Red-Blue Cherries position an integer? See also [10].
A6. (17) Extend the analysis of Kotzig’s Nim (WW, pp. 515–517). Is the game
eventually periodic in terms of the length of the circle for every finite move set?
Analyze the misère version of Kotzig’s Nim.
Let 0(S; n) be the outcome of the game S lists all the moves and n is the
size of the circle. Recently, Ward and Xin [2010] gave more evidence of the
periodicity by showing: If n ∈ {1, 3, 5, 7, 15}, or if n ≡ 3 (mod 5) and n ≥ 23,
then 0(1, 4; n) ∈ P; otherwise, 0(1, 4; n) ∈ N. They also give several conjectures
including: If n is odd and n 6∈ {9, 11, 17} then 0(3, 5; n) ∈ P and is in N otherwise.
Their evidence bolsters the periodicity conjecture but also indicates that there is
likely to be a lot of noise when n is small.
A7. (18) Obtain asymptotic estimates for the proportions of N-, O- and Ppositions in Epstein’s Put-or-Take-a-Square game (WW, pp. 518–520).
A8. Gale’s nim. This is Nim played with four heaps, but the game ends when
three of the heaps have vanished, so that there is a single heap left. Brouwer and
Guy have independently given a partial analysis, but the situation where the four
heaps have distinct sizes greater than 2 is open. An obvious generalization is to
play with h heaps and play finishes when k of them have vanished.
A9. Euclid’s Nim is played with a pair of positive integers, a move being to
diminish the larger by any multiple of the smaller. The winner is the player who
reduces a number to zero. Analyses have been given by Cole and Davie [1969],
Spitznagel [1973], Lengyel [2003], Collins [2005], Fraenkel [2005] and Nivasch [2006]. Lengyel [2003] reports that Schwartz first found that (a, b) is
the sequential sum of nim-heaps given by the normal continued fraction of a/b,
the last number in the continued fraction depends on the presence or absence
of fibonacci numbers. Gurvich [2007] shows that the nim-value, g + (a, b) for
the pair (a, b) in normal play is the same as the misère nim-value, g − (a, b)
286
RICHARD J. NOWAKOWSKI
50
40
30
20
10
0.2
0.4
0.6
0.8
1
Figure 2. Partizan Euclid: mean atomic weights for 0 < q < p < 101.
except for (a, b) = (k Fi , k Fi+1 ) where k > 0 and Fi is the i-th Fibonacci number.
In this case, g + (k Fi , k Fi+1 ) = 0 and g − (k Fi , k Fi+1 ) = 1 if i is even and the
values are reversed if i is odd. Aaron Siegel notes that this can be restated as:
Euclid is tame, and the fickle positions have the form (k Fi , k Fi+1 ). The latter
have genus 01 when i is even and 10 when i is odd.
Collins and Lengyel [392] play Euclid’s Nim with three integers and solve
some special cases.
In Partizan Euclid, from the position ( p, q), where p = kq + r , 0 ≤ r < q,
Left moves to (q, r ) and Right to (q, (k + 1)q − p). The outcome classes are
investigated in [12]. Determining the values seems hard. The atomic weights may
be easier — see Figure 2, where the mean atomic-weights for 0 < q < p ≤ 100
are given, where the coordinates (x, y) are x = q/ p and y is mean atomic-weight
of ( p, q).
A10. (20) D.U.D.E.N.E.Y (WW, pp. 521–523) is Nim, but with an upper bound,
Y, on the number of beans that may be taken, and with the restriction that a
player may not repeat his opponent’s last move. If Y is even, the analysis is easy.
Some advance in the analysis, when Y is odd, has been made by Marc Wallace,
Alex Fink and Kevin Saff.
We can, for example, extend the table of strings of pearls given in WW, p. 523,
with the values of Y in Table 1, which have the pure periods shown, where
D=Y + 2, E=Y + 1. The first entry corrects an error of 128r + 31 in WW.
It seems likely that the string for Y = 22k+1 + 22k − 1 has the simple period E
for all values of k. But there is some evidence to suggest that an analysis will
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
256r + 31
512r + 97
1024r + 103
128r + 119
1024r + 127
512r + 151
DEE
DDEDDDE
DE
DEE
DEEE
DDDEE
512r + 153
512r + 159
512r + 225
512r + 255
512r + 257
512r + 297
DEE
DEE
DDE
E
DDDDE
DDEDEDE
287
1024r + 415
512r + 425
512r + 487
1024r + 521
1024r + 607
1024r + 735
DEE
DE
DEE
DDDE
DDE
DEEE
Table 1. Strings of “pearls” in D.U.D.E.N.E.Y for values of Y of
various forms.
never be complete. Indeed, consider the following table showing the fraction,
among 2k cases, that remain undetermined:
k=
3 5 6 7
1
2
5
16
9
32
11
64
8
9
21
128
33
256
10 11
12
13
14
15
16
17
60
512
177
2048
304
4096
556
8192
974
16384
1576
32768
2763
65536
97
1024
fraction
Moreover, the periods of the pearl-strings appear to become arbitrarily long.
A11. (21) Schuhstrings is the same as D.U.D.E.N.E.Y, except that a deduction
of zero is also allowed, but cannot be immediately repeated (WW, pp. 523–
524). In Winning Ways it was stated that it was not known whether there is any
Schuhstring game in which three or more strings terminate simultaneously. Kevin
Saff has found three such strings (when the maximum deduction is Y = 3430,
the three strings of multiples of 2793, 3059, 3381 terminate simultaneously) and
he conjectures that there can be arbitrarily many such simultaneous terminations.
A12. (22) Analyze Dude, i.e., unbounded Dudeney, or Nim in which you are
not allowed to repeat your opponent’s last move.
Let [h 1 , h 2 , . . . , h k ; m], h i ≤ h i+1 , be the game with heaps of size h 1 through
h k , where m is the move just made and m = 0 denotes a starting position. Then
[5], the P-positions are
[(2s + 1)22 j ; (2s + 1)22 j ] for k = 1,
[(2s + 1)22 j , (2s + 1)22 j ; 1] for k = 2,
and for k ≥ 3 the heap sizes are arbitrary, the only condition being that the
previous move was 1. The nim-values do not seem to show an easily described
pattern.
A13. Nim with pass. David Gale suggested an analysis of Nim played with the
option of a single pass by either of the players, which may be made at any time
up to the penultimate move. It may not be made at the end of the game. Once
a player has passed, the game is as in ordinary Nim. The game ends when all
heaps have vanished. Morrison, Friedman and Landsberg [2011] have looked at
this game with their renormalization techniques.
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RICHARD J. NOWAKOWSKI
A14. Games with a Muller twist. In such games, each player specifies a condition on the set of options available to her opponent on his next move.
In Odd-or-Even Nim, for example, each player specifies the parity of the
opponent’s next move. This game was analyzed by Smith and Stănică [2002],
who propose several other such games which are still open (see also Gavel and
Strimling [2004]).
The game of Blocking Nim proceeds in exactly the same way as ordinary
Nim with N heaps, except that before a given player takes his turn, his opponent
is allowed to announce a block, (a1 , . . . a N ); i.e., for each pile of counters, he has
the option of specifying a positive number of counters which may not be removed
from that pile. Flammenkamp, Holshouser and Reiter [2003, 2004] give the Ppositions for three-heap Blocking Nim with an incomplete block containing only
one number, and ask for an analysis of this game with a block on just two heaps,
or on all three. There are corresponding questions for games with more than
three heaps. Larsson [2011] shows with two piles of counters where at most k −1
moves, for some fixed k, may be blocked off at each stage. Then the P-positions
are of the form {x, y}, where either |y − x| < k and y − x ≡ k − 1 (mod 2) or
x + y < k.
Let S be a set of positive integers. The complementary subtraction game Ŝ
is played on a heap where the last act of a move is say whether the next subtraction
is to be a number from S or from its complement. Horrocks and Trenton [2008]
introduced this variant. They analyzed heaps up to size 8000 the case where S
is the Fibonacci numbers without finding periodicity although there appears to
be regularity. They also ask about the case S = {x | x ≡ a or b (mod c)}. They
report that the nim-values appear chaotic. The set of (an , an + n) where a1 = 1
and an = mex{ai : i < n} form the P-positions of Wythoff’s game. We ask what
happens in this game if S = {an : n + 1, 2, 3, . . .}?
A15. (13) Misère caternary and octal games. Misère analysis has been revolutionized by Thane Plambeck and Aaron Siegel [2008] with their concept of the
misère quotient of a game, though the number of unsolved problems continues
to increase. See Section E13 for more theoretical questions and an explanation
of some of the concepts mentioned here.
Plambeck and Siegel ask the specific questions:
(1) The misère quotient of ·07 (Dawson’s Kayles) has order 638 at heap size 33.
Is it infinite at heap size 34? Even if the misère quotient is infinite at heap 34
then, by Rédei’s theorem [8, p. 142; 17], it must be isomorphic to a finitely
presented commutative monoid. Call this monoid D34 . Exhibit a monoid
presentation of D34 , and having done that, exhibit D35 , D36 , etc, and explain
what is going on in general. Given a set of games A, describe an algorithm
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
289
to determine whether the misère quotient of A is infinite. Much harder: if
the quotient is infinite, give an algorithm to compute a presentation for it.
(2) Give complete misère analyses for any of the (normal-play periodic) octal
games that show “algebraic-periodicity” in misère play. Some examples are
·54, ·261, ·355, ·357, ·516 and ·724. Give a precise definition of algebraic
periodicity and describe an algorithm for detecting and generalizing it. This
is a huge question: if such an algorithm exists, it would likely instantly give
solutions to at least a half-dozen unsolved 2- and 3-digit octals.
Plambeck also offers prizes of US $500.00 for a complete analysis of Dawson’s Chess, ·137 (alias Dawson’s Kayles, ·07); US $200.00 for the “wild
quaternary game”, ·3102; and US $25.00 each for ·3122, ·3123 and ·3312.
There is a website miseregames.org which contains thousands of misère
quotients for octal games.
Siegel notes that Dawson first proposed his problem in 1935, making it perhaps
the oldest open problem in combinatorial game theory. [Michael Albert offers
the alternative “Is chess a first player win?”] It may be of historical interest to
note that Dawson showed the problem to one of the present authors around 1947.
Fortunately, he forgot that Dawson proposed it as a losing game, was able to
analyze the normal play version, rediscover the Sprague–Grundy theory, and get
Conway interested in games.
A16. A16 is now E12.
A17. Nem and mnem. Mem is played with heaps of tokens. Remove any
number of tokens from any one heap. The number of tokens removed must be at
least as large as the number that were removed on the previous move from that
heap. Equivalently, a “heap” is a pair of integers (n, k), and a move is to any
pair (n − i, i), where k ≤ i ≤ n.
Mnem is exactly the same as Mem, with additional options: Either player
may add tokens to a heap instead of removing them. If adding tokens, the number
added must be strictly less than the number of tokens added or removed on the
previous move. If removing tokens, the number must be at least as large as the
number added or removed on the previous move. So a move from (n, k) is to
(n − i, i) for k ≤ i ≤ n, as in Mem, or to (n + i, i) for 1 ≤ i < k.
Conway and (Aaron) Siegel have investigated this game. They conjecture:
Every position in Mnem has finite nim value. They verified this experimentally
up to n = 10, 000. They also conjecture: For both Mem and Mnem, if k 2 ≥ n,
then (n, k) = bn/kc. Siegel reports that they have no idea how to prove either so
these may be very difficult problems.
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RICHARD J. NOWAKOWSKI
Figure 3. Jiang v. Rui, MSRI, July, 2000.
B. Pushing and placing pieces
B1. (5) Go is of particular interest, partly because of the loopiness induced by
the “ko” rule, and many problems involve general theory; see E4 and E5.
Elwyn writes:
I attach one region that has been studied intermittently over the past
several years. The region occurs in the southeast corner of the board
(Figure 3). At move 85 Black takes the ko at L6. What then is the
temperature at N4? This position is copied from the game Jiang and
Rui played at MSRI in July 2000. In 2001, Bill Spight and I worked
out a purported solution by hand, assuming either Black komaster or
White komaster. I’ve recently been trying to get that rather complicated
solution confirmed by GoExplorer, which would then presumably also
be able to calculate the dogmatic solution. I’ve been actively pursuing
this off and on for the past couple weeks, and haven’t gotten there yet.
Elwyn also writes:
Nakamura (GONC3) has shown how capturing races in Go can be
analyzed by treating liberties as combinatorial games. Like atomic
weights, when the values are integers, each player’s best move reduces
his opponent’s resources by one. The similarities between atomic
weights and Nakamura’s liberties are striking.
Theoretical problem: Either find a common formulation which includes much
or all of atomic weight theory and Nakamura’s theory of liberties, OR find some
significant differences.
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
291
Important practical applied problem: Extend Nakamura’s theory to include
other complications which often arise in Go, such as simple kos, either internal
and/or external.
B2. Woodpush (see LIP, pp. 214, 275) is a game that involves ko but that is
simpler than Go. Woodpush is played on a finite strip of squares. Each square is
empty or occupied by a black or white piece. A piece of the current player’s color
retreats: Left retreats to the left and Right to the right — to the next empty square,
or off the board if there is no empty square; except, if there is a contiguous
string containing an opponent’s piece then it can move in the opposite direction
pushing the string ahead of it. Pieces can be moved off the end of the strip.
Immediate repetition of a global board positions is not allowed. A “ko” threat
must be played first. For example:
Left
Right
Left
Right
L R R → L R R
→
L RR → ko-threat →
RR
Note that Right’s first move to L R R is illegal because it repeats the immediately
prior board position and Left’s second move to L R R is also illegal so he must
play a ko-threat. Also note that in L R R, Right never has to play a ko-threat
since he can always push with either of his two pieces — with Left moving first,
Left
Right
L R R → L R R
→
L RR
→ ko-threat
→ Right answers ko-threat
→ L R R
→
L R R
Berlekamp, Plambeck, Ottaway, Aaron Siegel and Spight (work in progress)
use top-down thermography to analyze the three piece positions. What about more
pieces? Cazenave and Nowakowski (this volume) show that the position L .L .R.R
is ±4 but that L .L ...R.R and L .L .....R.R are draws by superko (repetition of
the board position after more than 2 moves).
B3. (40) Chess. Noam Elkies [2002] has examined Dawson’s chess, but played
under usual Chess rules, so that capture is not obligatory.
He would still welcome progress with his conjecture that the value ∗k occurs
for all k in (ordinary chess) pawn endings on sufficiently large chessboards.
Thea van Roode has suggested Impartial Chess, in which the players may
make moves of either color. Checks need not be responded to and Kings may be
captured. The winner could be the first to promote a pawn.
B4. (30) Chess: King and Rook. Low and Stamp [2006] have given a strategy
in which White wins the King and Rook vs. King problem within an 11 × 9
region. Kanungo and Low [2007] show that with the initial position WK(1, 1),
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RICHARD J. NOWAKOWSKI
WR(x, y) and BK(a, b), where 1 < x < a, White has a winning strategy on the
(a + b + 3) × (a + b + 5) board.
B5. Nonattacking queens. Noon and Van Brummelen [2006] alternately place
queens on an n × n chessboard so that no queen attacks another. The winner is
the last queen placer. They give nim-values for boards of sizes 1 ≤ n ≤ 10 as
1121312310 and ask for the values of larger boards.
B6. (55) Amazons. Müller [14] has shown that the 5 × 5 game is a first player
win and asks about the 6 × 6 game.
B7. Phutball, more properly, Conway’s Philosopher’s Football, is usually
played on a Go board with positions (i, j), −9 ≤ i, j ≤ 9 and the ball starting at
(0,0). For the rules, see WW, pp. 752–755. The game is loopy (see Section E5
below), and Nowakowski, Ottaway and Siegel (see [19]) discovered positions
that contained tame cycles, i.e., cycles with only two strings, one each of Left
and Right moves. Aaron Siegel asks if there are positions in such combinatorial
games which are stoppers but contain a wild cycle, i.e., one which contains more
than one alternation between Left and Right moves. Demaine, Demaine and
Eppstein [2002] show that it is NP-complete to decide if a player can win on the
next move. Loosen (see directional phutball) raises the question of whether
there is any P-position.
Phlag Phutball is a variant played on an n×n board with the initial position of
the ball at (0, 0) except that now only the ball may occupy the positions (2i, 2 j)
with both coordinates even. This eliminates “tackling”, and is an extension
of one-dimensional Oddish Phutball, analyzed in Grossman and Nowakowski
[2002]. The (3, 2n + 1) board (i.e., (i, j), i = 0, 1, 2 and −n ≤ j ≤ n) is already
interesting and requires a different strategy from that appropriate to Oddish
Phutball.
Loosen [2008] introduces Directional Phutball, a nonloopy version, which
is also played on a grid. The ball starts in the bottom left corner with Left’s
goal-line is the right edge, Right’s is the top edge. Players can only jump toward
their opponent’s goal-line, and win by jumping on over that goal-line. Men can
only be placed ahead of the ball (i.e., in the positive quadrant with origin on
the ball). She notes that there is no P-position in this game and that making
an off-parity move (“poultry”) can be good. She asks for a complete analysis
of the 2 × n board. She shows that the atomic weights of the m × n board for
2 ≤ m, n, ≤ 4 is 0 and is 0, 1, 1, 2 for 2 × n for n = 5, 6, 7, 8 respectively.
B8. Hex. (LIP, pp. 264–265) Nash’s strategy stealing argument shows that Hex
is a first player win but few quantitative results are known. Arneson et al. report
that 8 × 8 is solved as are most openings on the 9 × 9.
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
293
Garikai Campbell [2004] asks:
(1) For each n, what is the shortest path on an n × n board with which the first
player can guarantee a win?
(2) What is the least number of moves that guarantees the first player a win?
Campbell showed that this is at least n on an n × n board. Peng et al. [2010]
show that it is 7 on the 5 × 5 board.
B9. (54) Fox and geese. Berlekamp and Siegel [19, Chapter 2] and WW, pp. 669–
710, “analysed the game fairly completely, relying in part on results obtained
using CGSuite.” On p. 710 of WW the following open problems are given.
(1) Define a position’s span as the maximum occupied row-rank minus its
minimum occupied row-rank. Then quantify and prove an assertion such as the
following: If the backfield is sufficiently large, and the span is sufficiently large,
and if the separation is sufficiently small, and if the Fox is neither already trapped
in a daggered position along the side of the board, nor immediately about to be
so trapped, then the Fox can escape and the value is off.
(2) Show that any formation of three Geese near the center of a very tall board
has a “critical rank” with the following property: If the northern Goose is far
above, and the Fox is far below, then the value of the position is either positive,
HOT, or off, according as the northern Goose is closer, equidistant, or further
from the critical rank than the Fox.
(3) Welton asks what happens if the Fox is empowered to retreat like a Bishop,
going back several squares at a time in a straight line? More generally, suppose
his straight-line retreating moves are confined to some specific set of sizes. Does
{1, 3}, which maintains parity, give him more or less advantage than {1, 2}?
(4) What happens if the number of Geese and board widths are changed?
In Aaron Siegel’s thesis there are several other questions:
(5) In the critical position, with Geese at (we use the algebraic Chess notation of
a, b, c, d, . . . for the files and 1, 2, 3, . . . , n for the ranks) (b, n), (d, n), (e, n−1),
(h, n − 1), and Fox at (c, n − 1), which has value 1 + 2−(n−8) on an n × 8 board
with n ≥ 8 in the usual game, is the value −2n + 11 for all n ≥ 6 when played
with “Ceylonese rules”? (Fox allowed two moves at each turn.)
(6) On an n × 4 board with n ≥ 5 and Geese at (b, n) and (c, n − 1) do all Fox
positions have value over? With the Geese on (b, n) and (d, n) are only other
values 0 at (c, n − 1) and {over | 0} at (b, n − 2) and (d, n − 2)?
(7) On an n × 6 board with n ≥ 8 and Geese at (b, n), (d, n) and (e, n − 1) do
the positions (a, n − 2k + 1), (c, n − 2k + 1), (e, n − 2k + 1), all have value 0,
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RICHARD J. NOWAKOWSKI
Figure 4. Sequences of hare and hounds positions.
and those at (b, n − 2k), (d, n − 2k), (f, n − 2k) all have value Star?˙ And if the
Geese are at (b, n), (d, n) and (f, n) are the zeroes and Stars interchanged?
B10. Hare and Hounds. Aaron Siegel asks if the sequences of positions of
increasing board length shown in Section B10 are, on the left, increasingly
hot, and, on the right, have arbitrarily large negative atomic weight. He also
conjectures that the starting position on a 6n + 5 × 3 board, for n > 0, has value
−(n − 1) + b, c |0k0 00 . . . 0
n
o
where there are 2n + 4 zeroes and slashes and b = 0, a 0, {0 | off} ,
c = 0 ↓[2] ∗|00
and a = {0, ↓[2] ∗ | 0, ↓[2] ∗}.
B11. (4) Domineering. (WW, pp. 119–122, 138–142; LIP pp. 1–7, 260). Extend
the analysis.
(Left and Right take turns to place dominoes on a checker-board. Left orients
her dominoes North–South and Right orients his East–West. Each domino exactly
covers two squares of the board and no two dominoes overlap. A player unable
to play loses.)
See Berlekamp [1988] and the second edition of WW, pp. 138–142, where
some new values are given. For example David Wolfe and Dan Calistrate have
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
295
Figure 5. A Domineering position of value ±2∗.
found the values (to within ‘-ish’, i.e., infinitesimally shifted) of 4 × 8, 5 × 6 and
6 × 6 boards. The value for a 5 × 7 board is
o
n o
1 n 3 1 3 3 5
3 1
2 4 − 2 , 2 − 2 , 2 − 1 − 1 − 3 − 1, 2 − 2 − 1 − 3
Lachmann, Moore and Rapaport [2002] discover who wins on rectangular, toroidal and cylindrical boards of widths 2, 3, 5 and 7, but do not find their values.
Bullock [3, p. 84] showed that 19 × 4, 21 × 4, 14 × 6 and 10 × 8 are wins for
Left and that 10 × 10 is a first player win.
Berlekamp notes that the value of a 2 × n board, for n even, is only known to
within “ish”, and that there are problems on 3 × n and 4 × n boards that are still
open.
Berlekamp asks, as a hard problem, to characterize all hot Domineering
positions to within “ish”. As a possibly easier problem he asks for a Domineering
position with a new temperature, i.e., one not occurring in Table 1 on GONC,
p. 477. Gabriel Drummond-Cole [2002] found values with temperatures between
1.5 and 2. Figure 5 shows a position of value ±2∗ and temperature 2.
Shankar and Sridharan [2005] have found many Domineering positions with
temperatures other than those shown in Table 1 on p. 477 of GONC. Blanco and
Fraenkel [2] have obtained partial results for the game of Tromineering, played
with trominoes in place of (or, alternatively, in addition to) dominoes.
B12. NoGo can be found under the name “Anti-Atari Go” at the Sensei Library
(see senseis.xmp.net/?AntiAtariGo) and was invented independently by Neil
McKay. On a Go-board (or on any graph) pieces are placed as in Go, the only
restriction is that every connected group, of each player, must be adjacent to at
least one empty intersection. Kyle Burke has shown that NoGo is NP-hard on a
graph. McKay, Nowakowski and (Angela) Siegel found positions of value 1, 12 ,
1 1
4 , 8 on a Go board. Using CGSuite, one can find the nimbers ∗, ∗2, ∗4, ∗8 on a
one-dimensional board. The position • · ◦ · • · ◦ . . . , where • is a black piece, ◦ is
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RICHARD J. NOWAKOWSKI
Figure 6. Starting position for “Icelandic” 1 × n Dots-and-Boxes.
a white piece and ‘ · ’ is an empty space, is equivalent to the octal game .6 (see
Section A2) which is not known to be periodic. We ask for the nim-dimension
of both the one- and two-dimensional board. Is there a limit to the denominator
of the fractions found on the two-dimensional board?
C. Playing with pencil and paper
C1. (51) Dots-and-Boxes. Elwyn Berlekamp asks for a complete theory of the
“Icelandic” 1 × n, i.e., with starting position as in Figure 6.
See Berlekamp’s book [2000] for more problems about this popular children’s
(and adults’) game (see also WW, pp. 541–584; LIP, pp. 21–28, 260).
C2. (25) Sprouts. Extend the analysis of this Conway–Paterson game in either
the normal or misère form. (WW, pp. 564–568).
(A move joins two spots, or a spot to itself by a curve which doesn’t meet any
other spot or previously drawn curve. When a curve is drawn, a new spot must
be placed on it. The valence of any spot must not exceed three.)
C3. (26) Sylver coinage. (WW, pp. 575–597): Extend the analysis.
(Players alternately name distinct positive integers, but may not name a number
which is a sum, with repetitions allowed, of previously named integers. Whoever
names 1 loses.) Sicherman [2002] contains the most recent information.
C4. (28) Extend Úlehla’s or Berlekamp’s analysis of von Neumann’s Game
from directed forests to directed acyclic graphs (WW, pp. 570–572; Úlehla
[1980]).
(Von Neumann’s game, or Hackendot, is played on one or more rooted trees.
The roots induce a direction, towards the root, on each edge. A move is to delete
a node, together with all nodes on the path to the root, and all edges incident
with those nodes. Any remaining subtrees are rooted by the nodes that were
adjacent to deleted nodes.)
C5. (43) Inverting Hackenbush. Thea van Roode has written a thesis [24]
investigating both this and Reversing Hackenbush, but there is plenty of room
for further analysis of both games.
In Inverting Hackenbush, when a player deletes an edge from a component,
the remainder of the component is replanted with the new root being the pruning
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
297
point of the deleted edge. In Reversing Hackenbush, the colors of the edges
are all changed after each deletion. Both games are hot, in contrast to Blue-Red
Hackenbush (WW, pp. 1–7; LIP, pp. 82, 88, 111–112, 212, 266) which is cold,
and Green Hackenbush (WW, pp. 189–196), which is tepid.
C6. (42) Beanstalk and Beans-Don’t-Talk are games invented respectively by
John Isbell and John Conway. See Guy [1986]. Beanstalk is played between
Jack and the Giant. The Giant chooses a positive integer, n 0 . Then Jack and the
Giant play alternately n 1 , n 2 , n 3 , . . . , according to the rule
n i+1 = n i /2 if n i is even,
= 3n i ± 1if n i is odd;
that is, if n i is even, there’s only one option, while if n i is odd there are just two.
The winner is the person moving to 1.
We still don’t know if there are any O-positions (positions of infinite remoteness).
C7. (63) The Erdős–Szekeres game [7] (and see [18]) was introduced by
Harary, Sagan and West [1985]. From a deck of cards labeled from 1 through n,
Alexander and Bridget alternately choose a card and append it to a sequence of
cards. The game ends when there is an ascending subsequence of a cards or a
descending subsequence of d cards.
The game appears to have a strong bias towards the first player. Albert et al.,
(this volume) show that for d = 2 and a ≤ n the outcome is N or P according as
n is odd or even, and is O (drawn) if n < a. They conjecture that for a ≥ d ≥ 3
and all sufficiently large n, it is N with both normal and misère play, and also
with normal play when played with the rationals in place of the first n integers.
They also suggest investigating the form of the game in which players take
turns naming pairs (i, πi ) subject to the constraint that the chosen values form
part of the graph of some permutation of {1, 2, . . . , n}.
D. Disturbing and destroying
D1. (27) Extend the analysis of Chomp (WW, pp. 598–599; LIP pp. 19, 46,
216).
David Gale offered $300.00 for the solution of the infinite 3D version where
the board is the set of all triples (x, y, z) of nonnegative integers, that is, the
lattice points in the positive octant of R3 . The problem is to decide whether it is
a win for the first or second player.
Chomp (Gale [1974]) is equivalent to Divisors (Schuh [1952]). Chomp is
easily solved for 2 × n arrays, Sun [2002], and indeed a recent result by Steven
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RICHARD J. NOWAKOWSKI
Byrnes [2003] shows that any poset game eventually displays periodic behavior
if it has two rows, and a fixed finite number of other elements. See also the
Fraenkel poset games mentioned near the end of Section A2.
Thus, most of the work in recent years has been on three-rowed Chomp. The
situation becomes quite complicated when a third row is added, see Zeilberger
[2001] and Brouwer et al. [2005]. A novel approach (renormalization) is taken
by Friedman and Landsberg (GONC3). They demonstrate that three-rowed
Chomp exhibits certain scaling and self-similarity patterns similar to chaotic
systems. Is there a deterministic proof that there is a unique winning move from
a 3 × n rectangle? The renormalization approach is based on nonlinear dynamics
techniques from physics; its results are highly suggestive but as of yet not fully
mathematically rigorous.
Transfinite Chomp has been investigated by Huddleston and Shurman [2002].
An open question is to calculate the nim value of the position ω×4 — they conjecture this to be ω · 2, but it could be as low as 46, or even uncomputable! Perhaps
the most fascinating open question in Transfinite Chomp is their Stratification
Conjecture, which states that if the number of elements taken in a move is < ωi ,
then the change in the nim-value is also < ωi .
D2. (33) Subset Take-away. Given a finite set, players alternately choose proper
subsets subject to the rule that once a subset has been chosen no proper subset
can be removed. Last player to move wins.
Many people play the dual, that is, a nonempty subset must be chosen and
no proper superset of this can be chosen. We discuss this version of the game
which now can be considered a poset game with the sets ordered by inclusion.
The (n; k) Subset Take-away game is played using all subsets of sizes 1
through k of a n-element set. In the (n; n) game one has the whole set (i.e., the
set of size n) as an option, so a strategy-stealing argument shows this must be a
first player win.
(1) Gale and Neyman [1982], in their original paper on the game, conjectured
that the winning move in the (n; n) game is to remove just the whole set.
This is equivalent to the statement that the (n; n −1) game is a second-player
win, which has been verified only for n ≤ 5.
(2) A stronger conjecture states that (n; k) is a second player win if and only if
k + 1 divides n. This was proved in the original paper only for k = 1 or 2.
See also Fraenkel and Scheinerman [1991].
D3. (39) Sowing or Mancala games. There appears to have been no advance
on the papers mentioned in MGONC, although we feel that this should be a
fruitful field of investigation at several different levels.
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299
D4. Annihilation Games. k-annihilation. Initially place tokens on some of
the vertices of a finite digraph. Denote by ρout (u) the outvalence of a vertex u.
A move consists of removing a token from some vertex u, and “complementing”
t := min(k, ρout (u)) (immediate) followers of u, say v1 , . . . , vt : if there is a
token on v j , remove it; if there is no token there, put one on it. The player
making the last move wins. If there is no last move, the outcome is a draw. For
k = 1, there is an O(n 6 ) algorithm for deciding whether any given position is in
P, N, or O; and for computing an optimal next move in the last 2 cases (Fraenkel
and Yesha [1982]). Fraenkel asks: Is there a polynomial algorithm for k > 1?
For an application of k-annihilation games to lexicodes; see Fraenkel and Rahat
[2003].
D5. Toppling Dominoes (LIP, pp. 110–112, 274) is played with a row of vertical
dominoes each of which is either blue or red. A player topples one of his/her
dominoes to the left or to the right.
See Fink et al [this volume] for the proofs that every number occurs exactly
once and is a palindrome; there are exactly n positions with value ∗n. Several
conjectures are listed but the most intriguing seems to be: if G is a palindrome
then G’s value appears uniquely.
There are several variants of Toppling Dominoes. If all the dominoes must
be toppled in the same direction then this is a Hackenbush string. Timber [15]
is an impartial version of Toppling Dominoes played on a directed graph. The
dominoes are on the edges. A player chooses a domino which is toppled in the
direction of the edge. The dominoes on incident edges are then toppled (regardless
of the underlying edge) and the process is iterated. If only outcomes are required
then only trees are interesting. In [15], an algorithm is given to determine the
outcome class of a tree, which surprisingly requires nim-values. Values, however,
appear to be difficult to determine. The normal and misère versions on a path are
related to Dyck paths and Catalan numbers; see Section E15 for more details.
The partizan version, imaginatively called Partizan Timber, has Left and Right
dominoes placed on the edges of a directed graph. Any Toppling Dominoes
position can be transformed into a Partizan Timber position by subdividing
each edge and directing the edge toward the new vertex. A similar algorithm
to that of Timber can be used to determine the outcome class of a tree. The
uniqueness of numbers doesn’t hold even on a path. In the other direction, are
there any values that do not occur in the game?
D6. Hanoi Stick-Up is played with the disks of the Towers of Hanoi puzzle,
starting with each disk in a separate stack. A move is to place one stack on top
of another such that the size of the bottom of the first stack is less than the size
of the top of the second; the two stacks then fuse (&) into one. The only relevant
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RICHARD J. NOWAKOWSKI
information about a stack are its top and bottom sizes, and it’s often possible
to collapse the labeling of positions: so, for instance, starting with 8 disks and
fusing 1&7 and 2&5,
we have stacks
0 1&7 2&5 3 4 6
which can be relabeled 0 1&3 1&2 1 2 3
in which the legal moves are still the same. John Conway, Alex Fink and others
have found that the P-positions of height ≤ 3 in normal Hanoi Stickup are
exactly those which, after collapsing, are of the form 0a 01b 1c 12d 2e with
min(a + b + c, c + d + e, a + e) even, except that if a + e ≤ a + b + c and
a + e ≤ c + d + e then both a and e must be even (02 can’t be involved in a legal
move so can be dropped).
They also found the normal and misère outcomes of all positions with up to
six stacks, but there is more to be discovered.
D7. (56) Are there any draws in Beggar-My-Neighbor? Marc Paulhus showed
that there are no cycles when using a half-deck of two suits, but the problem for
the whole deck (one of Conway’s “antiHilbert” problems) is still open.
E. Theory of games
E1. (49) Fraenkel updates Berlekamp’s questions on computational complexity
as follows:
Demaine, Demaine and Eppstein [2002] proved that deciding whether a player
can win in a single move in Phutball (WW, pp. 752–755; LIP, p. 212) is NPcomplete. Grossman and Nowakowski [2002] gave constructive partial strategies
for one-dimensional Phutball. Thus, these papers do not show that Phutball
has the required properties.
Perhaps Nimania (Fraenkel and Nešetřil [1985]) and Multivision (Fraenkel
[1998]) satisfy the requirements. Nimania begins with a single positive integer,
but after a while there is a multiset of positive integers on the table. At move
k, a copy of an existing integer m is selected, and 1 is subtracted from it. If
m = 1, the copy is deleted. Otherwise, k copies of m − 1 are adjoined to the
copy m − 1. The player first unable to move loses and the opponent wins. It was
proved: (i) The game terminates. (ii) Player I can win. In Fraenkel, Loebl and
Nešetřil [1988], it was shown that the max number of moves in Nimania is an
n−2
n−1
Ackermann function, and the min number satisfies 22 ≤ Min(n) ≤ 22 .
The game is thus intractable simply because of the length of its play. This is
a provable intractability, much stronger than NP-hardness, which is normally
only a conditional intractability. One of the requirements for the tractability of a
game is that a winner can consummate a win in at most O(cn ) moves, where
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
301
c > 1 is a constant, and n a sufficiently succinct encoding of the input (this much
is needed for nim on 2 equal heaps of size n).
To consummate a win in Nimania, player I can play randomly most of the
time, but near the end of play, a winning strategy is needed, given explicitly.
Whether or not this is an “intricate” solution, depends on the beholder. But it
seems that it’s of even greater interest to construct a game with a very simple
strategy which still has high complexity!
Also every play of Multivision terminates, the winner can be determined in
linear time, and the winning moves can be computed linearly. But the length of
play can be arbitrarily long. So let’s ask the following: Is there a game which
has
1. simple, playable rules,
2. a simple explicit strategy,
3. length of play at most exponential, and
4. is NP-hard or harder.
Tung [1987] proved the following:
Theorem. Given a polynomial P(x, y) ∈ Z[x, y], the problem of deciding
whether for all x there exists y[P(x, y) = 0] holds over Z≥0 , is co-NP-complete.
Define the following game of length 2: player I picks x ∈ Z≥0 , player II picks
y ∈ Z≥0 . Player I wins if P(x, y) 6= 0, otherwise player II wins. For winning,
player II has only to compute y such that P(x, y) = 0, given x, and there are
many algorithms for doing so.
Also Jones and Fraenkel [1995] produced games, with small length of play,
which satisfy these conditions.
So we are led to the following reformulation of Berlekamp’s question: Is there
a game which has
1. simple, playable rules,
2. a finite set of options at every move,
3. a simple explicit strategy,
4. length of play at most exponential,
5. and is NP-hard or harder.
E2. Complexity closure. Aviezri Fraenkel asks: Are there partizan games G 1 ,
G 2 , G 3 such that: (i) G 1 , G 2 , G 3 , G 1 + G 2 , G 2 + G 3 and all their options have
polynomial-time strategies, (ii) G 1 + G 3 is NP-hard?
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RICHARD J. NOWAKOWSKI
E3. Sums of switch games. David Wolfe considers a sum of games G, each
of the form akb|c or a|bkc where a, b, and c are integers specified in unary. Is
there a polynomial time algorithm to determine who wins in G, or is the problem
NP-hard?
E4. (52) How does one play sums of games with varied overheating operators?
Sentestrat and Top-down thermography (LIP, p. 214):
David Wolfe would like to see a formal proof that sentestrat works, an algorithm for top-down thermography, and conditions for which top-down thermography is computationally efficient.
Aaron Siegel asks the following generalized thermography questions.
(1) Show that the Left scaffold of a dogmatic (neutral ko-threat environment;
LIP, p. 215) thermograph is decreasing as function of t. (Note, this is NOT
true for komaster thermographs.) [Dogmatic thermography was invented by
Berlekamp and Spight. See [21] for a good introduction.]
(2) Develop the machinery for computing dogmatic thermographs of double
kos (multiple alternating 2-cycles joined at a single node).
In the same vein as (2):
(3) Develop a temperature theory that applies to all loopy games.
Siegel thinks that (3) is among the most important open problems in combinatorial game theory. The temperature theory of Go appears radically different
from the classical combinatorial theory of loopy games (where infinite plays
are draws). It would be a huge step forward if these could be reconciled into a
“grand unified temperature theory”. Problem (2) seems to be the obvious next
step toward (3).
R ∗ Conway asks for a natural set of conditions under which the mapping G 7→
G is the unique homomorphism that annihilates all infinitesimals.
E5. Loopy games (WW, pp. 334–377; LIP, pp. 213–214) are partizan games
that do not satisfy the ending condition. A Stopper is a game that, when played
on its own, has no ultimately alternating, Left and Right, infinite sequence of
legal moves. Aaron Siegel reminds us of WW, 2nd ed., p. 369, where the authors
tried hard to prove that every loopy game had stoppers, until Clive Bach found
the Carousel counterexample. Is there an alternative notion of simplest form
that works for all finite loopy games, and, in particular, for the Carousel? The
simplest form theorem for stoppers is at WW, p. 351.
Siegel conjectures that, if Q is an arbitrary cycle of Left and Right moves
that contains at least two moves for each player, and is not strictly alternating,
then there is a stopper consisting of a single cycle that matches Q, together
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
303
Figure 7. A Domineering position and a Chilled Go position of value ∗2.
with various exits to enders, i.e., games which end in a finite, though possibly
unbounded, number of moves. (Note that games normally have Left and Right
playing alternately, but if the game is a sum, then play in one component can
have arbitrary sequences of Left and Right moves, not just alternating ones.)
A long cycle is tame if it alternates just once between Left and Right, otherwise
it is wild. Aaron Siegel writes:
I can produce wild cycles “in the laboratory,” by specifying their game
graphs explicitly. So the question is to detect one “in nature”, i.e.,
in an actual game with (reasonably) playable rules such as phutball
[Problem B7].
Siegel also asks under what conditions does a given infinitesimal have a welldefined atomic weight, and asks to specify an algorithm to calculate the atomic
weight of an infinitesimal stopper g. The algorithm should succeed whenever
the atomic weight is well-defined, i.e., whenever g can be sandwiched between
loopfree all-smalls of equal atomic weight.
E6. (45) Elwyn Berlekamp asks for the habitat of ∗2, where ∗2 = {0, ∗|0, ∗}.
Gabriel Drummond-Cole [2005] has found Domineering positions with this
value. See, for example, Figure 7, which also shows a Go position, found by
Nakamura and Berlekamp [2003], whose chilled value is ∗2. The Black and
White groups are both connected to life via unshown connections emanating
upwards from the second row. Either player can move to ∗ by placing a stone
at E2, or to 0 by going to E1. Given a game, let n be the smallest nonnegative
integer n such that if a position of the game has value ∗k then k < 2n . Carlos
dos Santos calls n the nim-dimension of the game. He shows, this volume, that
Konane has infinite nim-dimension and ask for the nim-dimension of other
games.
E7. Partial ordering of games. David Wolfe lets g(n) be the number of games
born by day n, notes that an upper bound is given by g(n + 1) ≤ g(n) + 2g(n) + 2,
α
and a lower bound for each α < 0 is given by g(n + 1) ≥ 2g(n) , for n sufficiently
large, and asks us to tighten these bounds.
304
RICHARD J. NOWAKOWSKI
He also asks what group is generated by the all-small games (or — much
harder — of all games) born by day 3. Describe the partial order of games born
by day 3, identifying all the largest “hypercubes” (Boolean sublattices) and how
they are interconnected. These questions have been answered for day 2, see this
volume, pp.??.
Berlekamp suggests other possible definitions for games born by day n, Gn ,
depending on how one defines G0 . Our definition is 0-based, as G0 = {0}. Other
natural definitions are integer-based (where G0 are integers) or number-based.
These two alternatives do not form a lattice, for if G 1 and G 2 are born by day k,
then the games
Hn := G 1 , G 2 G 1 , {G 2 |−n}
form a decreasing sequence of games born by day k + 2 exceeding any game
G ≥ G 1 , G 2 , and the day k + 2 join of G 1 and G 2 cannot exist. What is the
structure of the partial order given by one of these alternative definitions of
birthday?
The set of all short games does not form a lattice, but Calistrate, Paulhus
and Wolfe [2002] have shown that the games born by day n form a distributive
lattice Ln under the usual partial order. They ask for a description of the exact
structure of L3 . Siegel describes L4 as “truly gigantic and exceedingly difficult
to penetrate” but suggests that it may be possible to find its dimension and the
maximum longitude, long4 (G), of a game in L4 , which he defines as
longn (G) = rankn (G ∨ G • ) − rankn (G),
where rankn (G) is the rank of G in Ln and G • is the companion of G,

∗
if G = 0,



{0, (G L )• | (G R )• } if G > 0,
G• =

{(G L )• | 0, (G R )• } if G < 0,



{(G L )• | (G R )• }
if G k 0.
Albert and Nowakowski [2011] show that starting with any set of games,
instead of just 0 = {·|·}, then the games born on the next day form a complete
but not necessarily distributive lattice. They ask which sets of games will give a
distributive lattice? Is there a set which will give a nondistributive but modular
lattice? Carvalho, dos Santos, Dias, Coelho, Neto, Nowakowski and Vinagre [4]
answered this by showing that for any lattice L, there is a set of games which
generates L on the next day.
The set of all-small games does not form a lattice, but Siegel forms a lattice
0
Ln by adjoining least and greatest elements 4 and 5 and asks: do the elements
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
305
of L0n have an intrinsic “handedness” that distinguishes, say, (n −1)·↑ from
(n−1)·↑ + ∗?
A game is option-closed (Nowakowski and Ottaway [2011]) if, recursively,
each G L L is also G L and the same for Right. For example, Hackenbush strings
are option-closed. Nowakowski and (Angela) Siegel [16] show that the optionclosed games born on day n form a planar, nondistributive lattice, but the question
of how many? remains unanswered. Are there other natural families of games
that form planar lattices?
E8. Aaron Siegel asks, given a group or monoid, K, of games, to specify a
technique for calculating the simplest game in each K-equivalence class. He
notes that some restriction on K might be needed; for example, K might be the
monoid of games absorbed by a given idempotent.
E9. Siegel also would like to investigate how search methods might be integrated
with a canonical-form engine.
E10. (9) Develop a misère theory for unions of partizan games (WW, p. 312).
E11. Four-outcome-games. Guy [2007] has given a brute force analysis of a
parity subtraction game which didn’t allow the use of Sprague–Grundy theory
because it wasn’t impartial, nor the Conway theory, because it was not lastplayer-winning. Is there a class of games in which there are four outcomes, Next,
Previous, Left and Right, and for which a general theory can be given?
E12. Impartial Misère Analysis. See A16. From the works of Plambeck and
Siegel: Let A be some set of games (the universe) played under misère rules.
Typically, A is the set of positions that arise in a particular game, such as
Dawson’s Chess. Games H, K ∈ A are said to be equivalent, denoted by
H ≡ K (modA, if H + X and K + X have the same outcome for all games X ∈ A.
The relation ≡ is an equivalence relation, and a set of representatives, one from
each equivalence class, forms the misère quotient, Q = A/≡. A quotient map
8 : A → Q is defined, for G ∈ A, by 8 : G = [G]≡ .
(1) A quotient map 8 : A → Q is said to be faithful if, whenever 8(G) = 8(H ),
then G and H have the same normal-play Grundy value. Is every quotient
map faithful?
(2) Let (Q, P) be a quotient and S a maximal subgroup of Q. Must S ∩ P be
nonempty? (Note: it’s easy to get a “yes” answer in the special case when
S is the kernel.)
(3) Extend the classification of impartial misère quotients. We have preliminary
results on the number of quotients of order n ≤ 18 but believe that this can
be pushed far higher.
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RICHARD J. NOWAKOWSKI
(4) Exhibit an impartial misère quotient with a period-5 element. Same question
for period 8, etc. We’ve detected quotients with elements of periods 1, 2,
3, 4, 6, and infinity, and we conjecture that there is no restriction on the
periods of quotient elements.
(5) In the flavor of both (3) and (4): What is the smallest quotient containing a
period 4 (or 3 or 6) element?
E13. Inverses in Partizan misère games. Siegel [Misère Canonical Forms of
Partizan Games, this volume] and Allen [this volume] extends the misère monoid
concepts to partizan games. In normal play, games in both players have a move
or neither does are called all-small because their values are infinitesimal. This
is not true in misère play and the term dicot has been coined to refer to these
games in all play conventions. (See Section E14 as well.)
Recall that −G is obtained from the game G by reversing the roles of Left
and Right — “turning the board around”. In normal play, G − G = 0 which is
decidedly not true in misère play. To avoid confusion, when dealing with misère
play −G is frequently written as G and is called the conjugate of G.
(1) Allen asks, in the dicot universe, D, when is it true that G + G ≡ 0(mod D)?
For example, ∗ + ∗ ≡ 0(mod D). McKay, Milley and Nowakowski [11]
show that this is true if G = ∗ : x where x is a number in canonical form
(regarded as a game tree) and more generally: if o− (H + H ) = N for every
H of G, including G, then G + G ≡ 0(mod D).
(2) Rebecca Milley asks, in a universe, U, closed under sums, conjugation and
subpositions if G + H ≡ 0(mod U) is it necessarily true that H = G? Jason
Brown had asked the question about arbitrary universes. Milley [13] has
the counterexample: on a finite strip of squares, Left places a 1 × 1 piece
and Right places a 1 × 2 (domino). In the universe of sums of strips a strip
of length 1 plus a strip of length 2 is equivalent to 0.
E14. Scoring games. Instead of “the last play determines the winner”, another
natural way is to have “whichever player has the higher score wins”. Both
Dots-and-Boxes and Go, for example, are scoring games.
Milnor looked at dicot scoring games with nonnegative incentives. Milnor
[1953] defines a “positional game” to be what we would call a dicot (Section E13)
scoring game, but only looks at positional games with nonnegative incentives.
Hanner [1959] looked at the same class of games as Milnor, and invented thermography. Ettinger [2000] looked at dicot scoring games (possibly with negative
incentives), which he calls “positional games” following Milnor. Johnson [9]
looked at the dicot scoring games that are “well-tempered” in some sense (a
subset of Ettinger’s games). Stewart [23; 22] looks at general scoring games.
UNSOLVED PROBLEMS IN COMBINATORIAL GAMES
307
Since the games are no longer dicot, a choice has to be made about how to handle
the situation where the current player can’t move but their opponent can.
There is much work yet to be done in this field. Johnson mentions one problem
worthy of a place to start: is there a general theory that covers Dots-and-Boxes
(Section C1)?
E15. Find a formula for the number of P-positions for Nim played with 2n
tokens. there are none with an odd number of tokens. The On-line Encyclopedia
of Integer Sequences (OEIS) has the value for small values of n.
In general, given a game, enumerate the number of P-positions of a given
size. We only know of Timber (Section D5) where the number of P-positions,
both normal and misère are related to Dyck paths with certain properties (also to
Catalan numbers and Fine numbers) and Heytei [2009; 2010] which relates the
number of P-positions to the Bernoulli numbers of the second kind.
Acknowledgement
I have had help in compiling this collection from all those mentioned, and from
others. We would especially like to mention Elwyn Berlekamp, Aviezri Fraenkel,
Will Johnson, Adam Landsberg, Urban Larsson, Neil McKay, Thane Plambeck,
Carlos Santos, Aaron Siegel and David Wolfe. All mistakes are deliberate and
designed to keep the reader alert.
References
[1] M. H. Albert, J.P Grossman, S. McCurdy, R. J. Nowakowski, and D. Wolfe. Cherries. preprint,
2005. [Problem A5].
[2] Saúl A. Blanco and Aviezri S. Fraenkel. Triomineering, tridomineering and l-tridomineering.
preprint, 2006. [Problem B11].
[3] N. Bullock. Domineering: solving large combinatorial search spaces. ICGA J., 25:67–85,
2002. [Problem B11].
[4] Alda Carvalho, Carlos Pereira dos Santos, Catia Dias, Francisco Coelho, Joao Pedro Neto,
Richard Nowakowski, and Sandra Vinagre. On lattices from combinatorial game theory
modularity and a representation theorem: finite case. Theoretical Computer Science, 527:37–
49, 2014. [Problem E7].
[5] N. Comeau, J. Cullis, R. J. Nowakowski, and J. Paek. Personal communication (class project).
[Problem A12].
[6] Carlos dos Santos. Alguns resultados sobre jogos imparciais e dimensao NIM. PhD thesis,
University of Lisbon, 2010. [Problem A1].
[7] Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio Math.,
2:463–470, 1935. [Problem C7].
[8] P. A. Grillet. Commutative semigroups. Number 2 in Advances in Mathematics (Dordrecht).
Kluwer Academic Publishers, Dordrecht, 2001. [Problem A15].
308
[9]
RICHARD J. NOWAKOWSKI
William Johnson. Well-tempered scoring games. Int. J. Game Theory, 43(2):415–438, 2014.
[Problem E14].
[10] Sarah McCurdy. Two combinatorial games. Master’s thesis, Dalhousie University, 2004.
[Problem A5].
[11] Neil A. McKay, Rebecca Milley, and Richard J. Nowakowski. Misère-play Hackenbush
sprigs. preprint, 2012. to appear in Intl. J. Game Theory, [Problem E13].
[12] Neil A. McKay and Richard J. Nowakowski. Outcomes of partizan Euclid. Integers, 12B:Paper
A9, 15, 2012/13. [Problem A9].
[13] Rebecca Milley. Restricted universes of partizan misère games. PhD thesis, Dalhousie
University, 2013. [Problem E13].
[14] M. Müller. Solving 5 × 5 amazons. In The 6th Game Programming Workshop, number 14 in
IPSJ Symposium Series, pages 64–71, 2001. [Problem B6].
[15] Richard J. Nowakowski, Gabriel Renault, Emily Lamoureux, Stephanie Mellon, and Timothy
Miller. The game of timber! J. of Combin. Math. and Combin. Comput., 85:213–225, 2013.
[16] Richard J. Nowakowski and Angela A. Siegel. The lattices of option-closed games. preprint.
[Problem E7].
[17] László Rédei. The theory of finitely generated commutative semigroups. Pergamon Press,
Oxford, 1965. [Problem A15].
[18] C. Schensted. Longest increasing and decreasing subsequences. Canad. J. Math., 13:179–191,
1961. Problem C7].
[19] Aaron Nathan Siegel. Loopy games and computation. PhD thesis, University of California,
Berkeley, Ann Arbor, MI, 2005. [Problem B7, B9].
[20] Angela Siegel. Finite excluded subtraction sets and infinite geography. Master’s thesis,
Dalhousie University, 2005. [Problem A1].
[21] W. L. Spight. Evaluating kos in a neutral threat environment: preliminary results. In Computers and games: third international conference, CG 2002, number 2883 in Lect. Notes in
Comput. Sci., pages 413–428. Springer, 2003. [Problem E4].
[22] Fraser Stewart. Impartial scoring games. preprint, 2012. Submitted, [Problem E14].
[23] Fraser Stewart. Scoring play combinatorial games. preprint, 2012. Submitted, [Problem E14].
[24] Thea van Roode. Partizan forms of Hackenbush. Master’s thesis, The University of Calgary,
2002. [Problem C5].
[email protected]
Department of Mathematics and Statistics,
Dalhousie University, Halifax, NS B3H 3J5, Canada